Problem: Three coplanar circles intersect as shown. What is the maximum number of points on the circles that a line passing through all three circles can touch?

[asy]import graph;
draw(Circle((-9,9),15));
draw(Circle((0,-9),15));
draw(Circle((9,9),15));
[/asy]
The maximum number of points at which a line can intersect 1 circle is 2 distinct points.  Thus, for 3 circles, the maximum should be $3 \times 2 = 6$ points at most.  If you're going for speed, you should probably guess 6 points at this point with a reasonable degree of certainty.  If you have time and want to be certain, you should only check for the existence of a line that intersects the three circles at $\boxed{6}$ distinct points, because it is impossible that a line could intersect the circles at more than 6 points.  (There are, in fact, many lines that satisfy the conditions.)